{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "# Quantization of Signals\n",
    "\n",
    "*This jupyter notebook is part of a [collection of notebooks](../index.ipynb) on various topics of Digital Signal Processing. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Oversampling\n",
    "\n",
    "[Oversampling](https://en.wikipedia.org/wiki/Oversampling) is a technique which is applied in [analog-to-digital converters](https://en.wikipedia.org/wiki/Analog-to-digital_converter) to lower the average power of the quantization error. It requires a joint consideration of sampling and quantization."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Ideal Analog-to-Digital Conversion\n",
    "\n",
    "Let's consider the ideal sampling of a signal followed by its quantization, as given by the following block diagram\n",
    "\n",
    "![Ideal analog to digital conversion of a signal](ideal_ADC.png)\n",
    "\n",
    "Ideal sampling is modeled by multiplying the continuous signal $x(t)$ with a series of equidistant Dirac impulse, resulting in the discrete signal $x[k] = x(k T)$ where $T$ denotes the sampling interval. The discrete signal $x[k]$ is then quantized. The output of the ideal analog-to-digital converter is the quantized discrete signal $x_\\text{Q}[k]$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Nyquist Sampling\n",
    "\n",
    "Sampling of the continuous signal $x(t)$ leads to repetitions of the spectrum $X(j \\omega) = \\mathcal{F} \\{ x(t) \\}$ at multiples of $\\omega_\\text{S} = \\frac{2 \\pi}{T}$. We limit ourselves to a continuous real-valued $x(t) \\in \\mathbb{R}$, band-limited signal $| X(j \\omega) | = 0$ for $|\\omega| > \\omega_\\text{C}$ where $\\omega_\\text{C}$ denotes its cut-off frequency. The spectral repetitions due to sampling do not overlap if the [sampling theorem](https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem) $\\omega_\\text{S} \\geq  2 \\cdot \\omega_\\text{C}$ is fulfilled. In the case of Nyquist (critical) sampling, the sampling frequency is chosen as $\\omega_\\text{S} =  2 \\cdot \\omega_\\text{C}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Oversampling\n",
    "\n",
    "The basic idea of oversampling is to sample the input signal at frequencies which are significantly higher than the Nyquist criterion dictates. After quantization, the signal is low-pass filtered by a discrete filter $H_\\text{LP}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ and resampled back to the Nyquist rate. In order to avoid aliasing due to the resampling this filter has to be chosen as an ideal low-pass\n",
    "\n",
    "\\begin{equation}\n",
    "H_\\text{LP}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\text{rect} \\left( \\frac{\\Omega}{2 \\, \\Omega_\\text{C}} \\right)\n",
    "\\end{equation}\n",
    "\n",
    "where $\\Omega_\\text{C} = \\omega_\\text{C} \\cdot T$. For an oversampling of factor $L \\in \\mathbb{N}$ we have $\\omega_\\text{S} = L \\cdot 2 \\omega_\\text{C}$. For this case, the resampling can be realized by keeping only every $L$-th sample which is known as decimation. The following block diagram illustrates the building blocks of oversampled digital-to-analog conversion, $\\downarrow L$ denotes decimation by a factor of $L$\n",
    "\n",
    "![Oversampling ideal analog to digital converter](oversampling.png)\n",
    "\n",
    "In order the conclude on the benefits of oversampling we have to derive the average power of the overall quantization error. According to our [model of the quantization error](linear_uniform_quantization_error.ipynb#Model-for-the-Quantization-Error), the quantization error $e[k]$ can be modeled as uniformly distributed white noise. Its power spectral density (PSD) is given as\n",
    "\n",
    "\\begin{equation}\n",
    "\\Phi_{ee}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega}) = \\frac{Q^2}{12}\n",
    "\\end{equation}\n",
    "\n",
    "where $Q$ denotes the quantization step. Before the discrete low-pass filter $H_\\text{LP}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$, the power of the quantization error is uniformly distributed over the entire frequency range $-\\pi < \\Omega \\leq \\pi$. However, after the ideal low-pass filter its frequency range is limited to $- \\frac{\\pi}{L} < \\Omega \\leq \\frac{\\pi}{L}$. The average power of the quantization error is then given as\n",
    "\n",
    "\\begin{equation}\n",
    "\\sigma_{e, \\text{LP}}^2 = \\frac{1}{2 \\pi} \\int\\limits_{- \\frac{\\pi}{L}}^{\\frac{\\pi}{L}} \\Phi_{ee}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})  \\; \\mathrm{d}\\Omega = \\frac{1}{L} \\cdot \\frac{Q^2}{12}\n",
    "\\end{equation}\n",
    "\n",
    "The average power $\\sigma_x^2$ of the sampled signal $x[k]$ is not affected, since the cutoff frequency of the low-pass filter has been chosen as the upper frequency limit $\\omega_\\text{C}$ of the input signal $x(t)$. \n",
    "\n",
    "In order to calculate the SNR of the oversampled analog-to-digital converter we assume that the input signal is drawn from a wide-sense stationary (WSS) uniformly distributed zero-mean random process with $x_\\text{min} \\leq x[k] < x_\\text{max}$. With the results from our discussion of [linear uniform quantization](linear_uniform_quantization_error.ipynb#Uniformly-Distributed-Signal) and $\\sigma_{e, \\text{LP}}^2$ from above we get\n",
    "\n",
    "\\begin{equation}\n",
    "SNR = 10 \\cdot \\log_{10} \\left( 2^{2 w} \\right) + 10 \\cdot \\log_{10} \\left( L \\right) \\approx 6.02 \\, w + 10 \\cdot \\log_{10} \\left( L \\right) \\quad \\text{in dB}\n",
    "\\end{equation}\n",
    "\n",
    "where $w$ denotes the number of bits used for a binary representation of the quantization index. Hence, oversampling by a factor of $L$ brings a plus of $10 \\cdot \\log_{10} \\left( L \\right)$ dB in terms of SNR. For instance, an oversampling by a factor of $L = 4$ results in a SNR which is approximately 6 dB higher. For equal SNR the quantization step $Q$ can be chosen larger. In terms of the wordlength of a quantizer this accounts to a reduction by one bit. Consequently, there is a trade-off between accuracy of the quantizer and its sampling frequency."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Example\n",
    "\n",
    "The following numerical simulation illustrates the benefit in terms of SNR for an oversampled linear uniform quantizer with $w = 16$ for the quantization of the harmonic signal $x[k] = \\cos[\\Omega_0 k]$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/pdf": "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\n",
      "image/svg+xml": [
       "<?xml version=\"1.0\" encoding=\"utf-8\" standalone=\"no\"?>\n",
       "<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.1//EN\"\n",
       "  \"http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd\">\n",
       "<!-- Created with matplotlib (https://matplotlib.org/) -->\n",
       "<svg height=\"266.538125pt\" version=\"1.1\" viewBox=\"0 0 612.165625 266.538125\" width=\"612.165625pt\" xmlns=\"http://www.w3.org/2000/svg\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">\n",
       " <metadata>\n",
       "  <rdf:RDF xmlns:cc=\"http://creativecommons.org/ns#\" xmlns:dc=\"http://purl.org/dc/elements/1.1/\" xmlns:rdf=\"http://www.w3.org/1999/02/22-rdf-syntax-ns#\">\n",
       "   <cc:Work>\n",
       "    <dc:type rdf:resource=\"http://purl.org/dc/dcmitype/StillImage\"/>\n",
       "    <dc:date>2021-01-18T12:27:48.118206</dc:date>\n",
       "    <dc:format>image/svg+xml</dc:format>\n",
       "    <dc:creator>\n",
       "     <cc:Agent>\n",
       "      <dc:title>Matplotlib v3.3.2, https://matplotlib.org/</dc:title>\n",
       "     </cc:Agent>\n",
       "    </dc:creator>\n",
       "   </cc:Work>\n",
       "  </rdf:RDF>\n",
       " </metadata>\n",
       " <defs>\n",
       "  <style type=\"text/css\">*{stroke-linecap:butt;stroke-linejoin:round;}</style>\n",
       " </defs>\n",
       " <g id=\"figure_1\">\n",
       "  <g id=\"patch_1\">\n",
       "   <path d=\"M 0 266.538125 \n",
       "L 612.165625 266.538125 \n",
       "L 612.165625 0 \n",
       "L 0 0 \n",
       "z\n",
       "\" style=\"fill:#ffffff;\"/>\n",
       "  </g>\n",
       "  <g id=\"axes_1\">\n",
       "   <g id=\"patch_2\">\n",
       "    <path d=\"M 46.965625 228.96 \n",
       "L 604.965625 228.96 \n",
       "L 604.965625 7.2 \n",
       "L 46.965625 7.2 \n",
       "z\n",
       "\" style=\"fill:#ffffff;\"/>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_1\">\n",
       "    <g id=\"xtick_1\">\n",
       "     <g id=\"line2d_1\">\n",
       "      <path clip-path=\"url(#p8f4f71aab9)\" d=\"M 219.560611 228.96 \n",
       "L 219.560611 7.2 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_2\">\n",
       "      <defs>\n",
       "       <path d=\"M 0 0 \n",
       "L 0 3.5 \n",
       "\" id=\"m0b8095e6b1\" style=\"stroke:#000000;stroke-width:0.8;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"219.560611\" xlink:href=\"#m0b8095e6b1\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_1\">\n",
       "      <!-- $\\mathdefault{10^{1}}$ -->\n",
       "      <g transform=\"translate(210.760611 243.558437)scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path d=\"M 12.40625 8.296875 \n",
       "L 28.515625 8.296875 \n",
       "L 28.515625 63.921875 \n",
       "L 10.984375 60.40625 \n",
       "L 10.984375 69.390625 \n",
       "L 28.421875 72.90625 \n",
       "L 38.28125 72.90625 \n",
       "L 38.28125 8.296875 \n",
       "L 54.390625 8.296875 \n",
       "L 54.390625 0 \n",
       "L 12.40625 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-49\"/>\n",
       "        <path d=\"M 31.78125 66.40625 \n",
       "Q 24.171875 66.40625 20.328125 58.90625 \n",
       "Q 16.5 51.421875 16.5 36.375 \n",
       "Q 16.5 21.390625 20.328125 13.890625 \n",
       "Q 24.171875 6.390625 31.78125 6.390625 \n",
       "Q 39.453125 6.390625 43.28125 13.890625 \n",
       "Q 47.125 21.390625 47.125 36.375 \n",
       "Q 47.125 51.421875 43.28125 58.90625 \n",
       "Q 39.453125 66.40625 31.78125 66.40625 \n",
       "z\n",
       "M 31.78125 74.21875 \n",
       "Q 44.046875 74.21875 50.515625 64.515625 \n",
       "Q 56.984375 54.828125 56.984375 36.375 \n",
       "Q 56.984375 17.96875 50.515625 8.265625 \n",
       "Q 44.046875 -1.421875 31.78125 -1.421875 \n",
       "Q 19.53125 -1.421875 13.0625 8.265625 \n",
       "Q 6.59375 17.96875 6.59375 36.375 \n",
       "Q 6.59375 54.828125 13.0625 64.515625 \n",
       "Q 19.53125 74.21875 31.78125 74.21875 \n",
       "z\n",
       "\" id=\"DejaVuSans-48\"/>\n",
       "       </defs>\n",
       "       <use transform=\"translate(0 0.684375)\" xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use transform=\"translate(63.623047 0.684375)\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use transform=\"translate(128.203125 38.965625)scale(0.7)\" xlink:href=\"#DejaVuSans-49\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_2\">\n",
       "     <g id=\"line2d_3\">\n",
       "      <path clip-path=\"url(#p8f4f71aab9)\" d=\"M 430.201052 228.96 \n",
       "L 430.201052 7.2 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_4\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"430.201052\" xlink:href=\"#m0b8095e6b1\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_2\">\n",
       "      <!-- $\\mathdefault{10^{2}}$ -->\n",
       "      <g transform=\"translate(421.401052 243.558437)scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path d=\"M 19.1875 8.296875 \n",
       "L 53.609375 8.296875 \n",
       "L 53.609375 0 \n",
       "L 7.328125 0 \n",
       "L 7.328125 8.296875 \n",
       "Q 12.9375 14.109375 22.625 23.890625 \n",
       "Q 32.328125 33.6875 34.8125 36.53125 \n",
       "Q 39.546875 41.84375 41.421875 45.53125 \n",
       "Q 43.3125 49.21875 43.3125 52.78125 \n",
       "Q 43.3125 58.59375 39.234375 62.25 \n",
       "Q 35.15625 65.921875 28.609375 65.921875 \n",
       "Q 23.96875 65.921875 18.8125 64.3125 \n",
       "Q 13.671875 62.703125 7.8125 59.421875 \n",
       "L 7.8125 69.390625 \n",
       "Q 13.765625 71.78125 18.9375 73 \n",
       "Q 24.125 74.21875 28.421875 74.21875 \n",
       "Q 39.75 74.21875 46.484375 68.546875 \n",
       "Q 53.21875 62.890625 53.21875 53.421875 \n",
       "Q 53.21875 48.921875 51.53125 44.890625 \n",
       "Q 49.859375 40.875 45.40625 35.40625 \n",
       "Q 44.1875 33.984375 37.640625 27.21875 \n",
       "Q 31.109375 20.453125 19.1875 8.296875 \n",
       "z\n",
       "\" id=\"DejaVuSans-50\"/>\n",
       "       </defs>\n",
       "       <use transform=\"translate(0 0.765625)\" xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use transform=\"translate(63.623047 0.765625)\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use transform=\"translate(128.203125 39.046875)scale(0.7)\" xlink:href=\"#DejaVuSans-50\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_3\">\n",
       "     <g id=\"line2d_5\">\n",
       "      <defs>\n",
       "       <path d=\"M 0 0 \n",
       "L 0 2 \n",
       "\" id=\"m1fea7a5ba5\" style=\"stroke:#000000;stroke-width:0.6;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"72.329261\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_4\">\n",
       "     <g id=\"line2d_6\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"109.421202\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_5\">\n",
       "     <g id=\"line2d_7\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"135.738352\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_6\">\n",
       "     <g id=\"line2d_8\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"156.15152\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_7\">\n",
       "     <g id=\"line2d_9\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"172.830293\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_8\">\n",
       "     <g id=\"line2d_10\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"186.931994\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_9\">\n",
       "     <g id=\"line2d_11\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"199.147443\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_10\">\n",
       "     <g id=\"line2d_12\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"209.922233\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_11\">\n",
       "     <g id=\"line2d_13\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"282.969702\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_12\">\n",
       "     <g id=\"line2d_14\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"320.061642\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_13\">\n",
       "     <g id=\"line2d_15\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"346.378793\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_14\">\n",
       "     <g id=\"line2d_16\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"366.791961\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_15\">\n",
       "     <g id=\"line2d_17\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"383.470733\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_16\">\n",
       "     <g id=\"line2d_18\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"397.572434\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_17\">\n",
       "     <g id=\"line2d_19\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"409.787884\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_18\">\n",
       "     <g id=\"line2d_20\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"420.562674\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_19\">\n",
       "     <g id=\"line2d_21\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"493.610142\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_20\">\n",
       "     <g id=\"line2d_22\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"530.702083\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_21\">\n",
       "     <g id=\"line2d_23\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"557.019233\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_22\">\n",
       "     <g id=\"line2d_24\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"577.432401\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"xtick_23\">\n",
       "     <g id=\"line2d_25\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.6;\" x=\"594.111174\" xlink:href=\"#m1fea7a5ba5\" y=\"228.96\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_3\">\n",
       "     <!-- oversampling factor $L$ -->\n",
       "     <g transform=\"translate(271.365625 257.238125)scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path d=\"M 30.609375 48.390625 \n",
       "Q 23.390625 48.390625 19.1875 42.75 \n",
       "Q 14.984375 37.109375 14.984375 27.296875 \n",
       "Q 14.984375 17.484375 19.15625 11.84375 \n",
       "Q 23.34375 6.203125 30.609375 6.203125 \n",
       "Q 37.796875 6.203125 41.984375 11.859375 \n",
       "Q 46.1875 17.53125 46.1875 27.296875 \n",
       "Q 46.1875 37.015625 41.984375 42.703125 \n",
       "Q 37.796875 48.390625 30.609375 48.390625 \n",
       "z\n",
       "M 30.609375 56 \n",
       "Q 42.328125 56 49.015625 48.375 \n",
       "Q 55.71875 40.765625 55.71875 27.296875 \n",
       "Q 55.71875 13.875 49.015625 6.21875 \n",
       "Q 42.328125 -1.421875 30.609375 -1.421875 \n",
       "Q 18.84375 -1.421875 12.171875 6.21875 \n",
       "Q 5.515625 13.875 5.515625 27.296875 \n",
       "Q 5.515625 40.765625 12.171875 48.375 \n",
       "Q 18.84375 56 30.609375 56 \n",
       "z\n",
       "\" id=\"DejaVuSans-111\"/>\n",
       "       <path d=\"M 2.984375 54.6875 \n",
       "L 12.5 54.6875 \n",
       "L 29.59375 8.796875 \n",
       "L 46.6875 54.6875 \n",
       "L 56.203125 54.6875 \n",
       "L 35.6875 0 \n",
       "L 23.484375 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-118\"/>\n",
       "       <path d=\"M 56.203125 29.59375 \n",
       "L 56.203125 25.203125 \n",
       "L 14.890625 25.203125 \n",
       "Q 15.484375 15.921875 20.484375 11.0625 \n",
       "Q 25.484375 6.203125 34.421875 6.203125 \n",
       "Q 39.59375 6.203125 44.453125 7.46875 \n",
       "Q 49.3125 8.734375 54.109375 11.28125 \n",
       "L 54.109375 2.78125 \n",
       "Q 49.265625 0.734375 44.1875 -0.34375 \n",
       "Q 39.109375 -1.421875 33.890625 -1.421875 \n",
       "Q 20.796875 -1.421875 13.15625 6.1875 \n",
       "Q 5.515625 13.8125 5.515625 26.8125 \n",
       "Q 5.515625 40.234375 12.765625 48.109375 \n",
       "Q 20.015625 56 32.328125 56 \n",
       "Q 43.359375 56 49.78125 48.890625 \n",
       "Q 56.203125 41.796875 56.203125 29.59375 \n",
       "z\n",
       "M 47.21875 32.234375 \n",
       "Q 47.125 39.59375 43.09375 43.984375 \n",
       "Q 39.0625 48.390625 32.421875 48.390625 \n",
       "Q 24.90625 48.390625 20.390625 44.140625 \n",
       "Q 15.875 39.890625 15.1875 32.171875 \n",
       "z\n",
       "\" id=\"DejaVuSans-101\"/>\n",
       "       <path d=\"M 41.109375 46.296875 \n",
       "Q 39.59375 47.171875 37.8125 47.578125 \n",
       "Q 36.03125 48 33.890625 48 \n",
       "Q 26.265625 48 22.1875 43.046875 \n",
       "Q 18.109375 38.09375 18.109375 28.8125 \n",
       "L 18.109375 0 \n",
       "L 9.078125 0 \n",
       "L 9.078125 54.6875 \n",
       "L 18.109375 54.6875 \n",
       "L 18.109375 46.1875 \n",
       "Q 20.953125 51.171875 25.484375 53.578125 \n",
       "Q 30.03125 56 36.53125 56 \n",
       "Q 37.453125 56 38.578125 55.875 \n",
       "Q 39.703125 55.765625 41.0625 55.515625 \n",
       "z\n",
       "\" id=\"DejaVuSans-114\"/>\n",
       "       <path d=\"M 44.28125 53.078125 \n",
       "L 44.28125 44.578125 \n",
       "Q 40.484375 46.53125 36.375 47.5 \n",
       "Q 32.28125 48.484375 27.875 48.484375 \n",
       "Q 21.1875 48.484375 17.84375 46.4375 \n",
       "Q 14.5 44.390625 14.5 40.28125 \n",
       "Q 14.5 37.15625 16.890625 35.375 \n",
       "Q 19.28125 33.59375 26.515625 31.984375 \n",
       "L 29.59375 31.296875 \n",
       "Q 39.15625 29.25 43.1875 25.515625 \n",
       "Q 47.21875 21.78125 47.21875 15.09375 \n",
       "Q 47.21875 7.46875 41.1875 3.015625 \n",
       "Q 35.15625 -1.421875 24.609375 -1.421875 \n",
       "Q 20.21875 -1.421875 15.453125 -0.5625 \n",
       "Q 10.6875 0.296875 5.421875 2 \n",
       "L 5.421875 11.28125 \n",
       "Q 10.40625 8.6875 15.234375 7.390625 \n",
       "Q 20.0625 6.109375 24.8125 6.109375 \n",
       "Q 31.15625 6.109375 34.5625 8.28125 \n",
       "Q 37.984375 10.453125 37.984375 14.40625 \n",
       "Q 37.984375 18.0625 35.515625 20.015625 \n",
       "Q 33.0625 21.96875 24.703125 23.78125 \n",
       "L 21.578125 24.515625 \n",
       "Q 13.234375 26.265625 9.515625 29.90625 \n",
       "Q 5.8125 33.546875 5.8125 39.890625 \n",
       "Q 5.8125 47.609375 11.28125 51.796875 \n",
       "Q 16.75 56 26.8125 56 \n",
       "Q 31.78125 56 36.171875 55.265625 \n",
       "Q 40.578125 54.546875 44.28125 53.078125 \n",
       "z\n",
       "\" id=\"DejaVuSans-115\"/>\n",
       "       <path d=\"M 34.28125 27.484375 \n",
       "Q 23.390625 27.484375 19.1875 25 \n",
       "Q 14.984375 22.515625 14.984375 16.5 \n",
       "Q 14.984375 11.71875 18.140625 8.90625 \n",
       "Q 21.296875 6.109375 26.703125 6.109375 \n",
       "Q 34.1875 6.109375 38.703125 11.40625 \n",
       "Q 43.21875 16.703125 43.21875 25.484375 \n",
       "L 43.21875 27.484375 \n",
       "z\n",
       "M 52.203125 31.203125 \n",
       "L 52.203125 0 \n",
       "L 43.21875 0 \n",
       "L 43.21875 8.296875 \n",
       "Q 40.140625 3.328125 35.546875 0.953125 \n",
       "Q 30.953125 -1.421875 24.3125 -1.421875 \n",
       "Q 15.921875 -1.421875 10.953125 3.296875 \n",
       "Q 6 8.015625 6 15.921875 \n",
       "Q 6 25.140625 12.171875 29.828125 \n",
       "Q 18.359375 34.515625 30.609375 34.515625 \n",
       "L 43.21875 34.515625 \n",
       "L 43.21875 35.40625 \n",
       "Q 43.21875 41.609375 39.140625 45 \n",
       "Q 35.0625 48.390625 27.6875 48.390625 \n",
       "Q 23 48.390625 18.546875 47.265625 \n",
       "Q 14.109375 46.140625 10.015625 43.890625 \n",
       "L 10.015625 52.203125 \n",
       "Q 14.9375 54.109375 19.578125 55.046875 \n",
       "Q 24.21875 56 28.609375 56 \n",
       "Q 40.484375 56 46.34375 49.84375 \n",
       "Q 52.203125 43.703125 52.203125 31.203125 \n",
       "z\n",
       "\" id=\"DejaVuSans-97\"/>\n",
       "       <path d=\"M 52 44.1875 \n",
       "Q 55.375 50.25 60.0625 53.125 \n",
       "Q 64.75 56 71.09375 56 \n",
       "Q 79.640625 56 84.28125 50.015625 \n",
       "Q 88.921875 44.046875 88.921875 33.015625 \n",
       "L 88.921875 0 \n",
       "L 79.890625 0 \n",
       "L 79.890625 32.71875 \n",
       "Q 79.890625 40.578125 77.09375 44.375 \n",
       "Q 74.3125 48.1875 68.609375 48.1875 \n",
       "Q 61.625 48.1875 57.5625 43.546875 \n",
       "Q 53.515625 38.921875 53.515625 30.90625 \n",
       "L 53.515625 0 \n",
       "L 44.484375 0 \n",
       "L 44.484375 32.71875 \n",
       "Q 44.484375 40.625 41.703125 44.40625 \n",
       "Q 38.921875 48.1875 33.109375 48.1875 \n",
       "Q 26.21875 48.1875 22.15625 43.53125 \n",
       "Q 18.109375 38.875 18.109375 30.90625 \n",
       "L 18.109375 0 \n",
       "L 9.078125 0 \n",
       "L 9.078125 54.6875 \n",
       "L 18.109375 54.6875 \n",
       "L 18.109375 46.1875 \n",
       "Q 21.1875 51.21875 25.484375 53.609375 \n",
       "Q 29.78125 56 35.6875 56 \n",
       "Q 41.65625 56 45.828125 52.96875 \n",
       "Q 50 49.953125 52 44.1875 \n",
       "z\n",
       "\" id=\"DejaVuSans-109\"/>\n",
       "       <path d=\"M 18.109375 8.203125 \n",
       "L 18.109375 -20.796875 \n",
       "L 9.078125 -20.796875 \n",
       "L 9.078125 54.6875 \n",
       "L 18.109375 54.6875 \n",
       "L 18.109375 46.390625 \n",
       "Q 20.953125 51.265625 25.265625 53.625 \n",
       "Q 29.59375 56 35.59375 56 \n",
       "Q 45.5625 56 51.78125 48.09375 \n",
       "Q 58.015625 40.1875 58.015625 27.296875 \n",
       "Q 58.015625 14.40625 51.78125 6.484375 \n",
       "Q 45.5625 -1.421875 35.59375 -1.421875 \n",
       "Q 29.59375 -1.421875 25.265625 0.953125 \n",
       "Q 20.953125 3.328125 18.109375 8.203125 \n",
       "z\n",
       "M 48.6875 27.296875 \n",
       "Q 48.6875 37.203125 44.609375 42.84375 \n",
       "Q 40.53125 48.484375 33.40625 48.484375 \n",
       "Q 26.265625 48.484375 22.1875 42.84375 \n",
       "Q 18.109375 37.203125 18.109375 27.296875 \n",
       "Q 18.109375 17.390625 22.1875 11.75 \n",
       "Q 26.265625 6.109375 33.40625 6.109375 \n",
       "Q 40.53125 6.109375 44.609375 11.75 \n",
       "Q 48.6875 17.390625 48.6875 27.296875 \n",
       "z\n",
       "\" id=\"DejaVuSans-112\"/>\n",
       "       <path d=\"M 9.421875 75.984375 \n",
       "L 18.40625 75.984375 \n",
       "L 18.40625 0 \n",
       "L 9.421875 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-108\"/>\n",
       "       <path d=\"M 9.421875 54.6875 \n",
       "L 18.40625 54.6875 \n",
       "L 18.40625 0 \n",
       "L 9.421875 0 \n",
       "z\n",
       "M 9.421875 75.984375 \n",
       "L 18.40625 75.984375 \n",
       "L 18.40625 64.59375 \n",
       "L 9.421875 64.59375 \n",
       "z\n",
       "\" id=\"DejaVuSans-105\"/>\n",
       "       <path d=\"M 54.890625 33.015625 \n",
       "L 54.890625 0 \n",
       "L 45.90625 0 \n",
       "L 45.90625 32.71875 \n",
       "Q 45.90625 40.484375 42.875 44.328125 \n",
       "Q 39.84375 48.1875 33.796875 48.1875 \n",
       "Q 26.515625 48.1875 22.3125 43.546875 \n",
       "Q 18.109375 38.921875 18.109375 30.90625 \n",
       "L 18.109375 0 \n",
       "L 9.078125 0 \n",
       "L 9.078125 54.6875 \n",
       "L 18.109375 54.6875 \n",
       "L 18.109375 46.1875 \n",
       "Q 21.34375 51.125 25.703125 53.5625 \n",
       "Q 30.078125 56 35.796875 56 \n",
       "Q 45.21875 56 50.046875 50.171875 \n",
       "Q 54.890625 44.34375 54.890625 33.015625 \n",
       "z\n",
       "\" id=\"DejaVuSans-110\"/>\n",
       "       <path d=\"M 45.40625 27.984375 \n",
       "Q 45.40625 37.75 41.375 43.109375 \n",
       "Q 37.359375 48.484375 30.078125 48.484375 \n",
       "Q 22.859375 48.484375 18.828125 43.109375 \n",
       "Q 14.796875 37.75 14.796875 27.984375 \n",
       "Q 14.796875 18.265625 18.828125 12.890625 \n",
       "Q 22.859375 7.515625 30.078125 7.515625 \n",
       "Q 37.359375 7.515625 41.375 12.890625 \n",
       "Q 45.40625 18.265625 45.40625 27.984375 \n",
       "z\n",
       "M 54.390625 6.78125 \n",
       "Q 54.390625 -7.171875 48.1875 -13.984375 \n",
       "Q 42 -20.796875 29.203125 -20.796875 \n",
       "Q 24.46875 -20.796875 20.265625 -20.09375 \n",
       "Q 16.0625 -19.390625 12.109375 -17.921875 \n",
       "L 12.109375 -9.1875 \n",
       "Q 16.0625 -11.328125 19.921875 -12.34375 \n",
       "Q 23.78125 -13.375 27.78125 -13.375 \n",
       "Q 36.625 -13.375 41.015625 -8.765625 \n",
       "Q 45.40625 -4.15625 45.40625 5.171875 \n",
       "L 45.40625 9.625 \n",
       "Q 42.625 4.78125 38.28125 2.390625 \n",
       "Q 33.9375 0 27.875 0 \n",
       "Q 17.828125 0 11.671875 7.65625 \n",
       "Q 5.515625 15.328125 5.515625 27.984375 \n",
       "Q 5.515625 40.671875 11.671875 48.328125 \n",
       "Q 17.828125 56 27.875 56 \n",
       "Q 33.9375 56 38.28125 53.609375 \n",
       "Q 42.625 51.21875 45.40625 46.390625 \n",
       "L 45.40625 54.6875 \n",
       "L 54.390625 54.6875 \n",
       "z\n",
       "\" id=\"DejaVuSans-103\"/>\n",
       "       <path id=\"DejaVuSans-32\"/>\n",
       "       <path d=\"M 37.109375 75.984375 \n",
       "L 37.109375 68.5 \n",
       "L 28.515625 68.5 \n",
       "Q 23.6875 68.5 21.796875 66.546875 \n",
       "Q 19.921875 64.59375 19.921875 59.515625 \n",
       "L 19.921875 54.6875 \n",
       "L 34.71875 54.6875 \n",
       "L 34.71875 47.703125 \n",
       "L 19.921875 47.703125 \n",
       "L 19.921875 0 \n",
       "L 10.890625 0 \n",
       "L 10.890625 47.703125 \n",
       "L 2.296875 47.703125 \n",
       "L 2.296875 54.6875 \n",
       "L 10.890625 54.6875 \n",
       "L 10.890625 58.5 \n",
       "Q 10.890625 67.625 15.140625 71.796875 \n",
       "Q 19.390625 75.984375 28.609375 75.984375 \n",
       "z\n",
       "\" id=\"DejaVuSans-102\"/>\n",
       "       <path d=\"M 48.78125 52.59375 \n",
       "L 48.78125 44.1875 \n",
       "Q 44.96875 46.296875 41.140625 47.34375 \n",
       "Q 37.3125 48.390625 33.40625 48.390625 \n",
       "Q 24.65625 48.390625 19.8125 42.84375 \n",
       "Q 14.984375 37.3125 14.984375 27.296875 \n",
       "Q 14.984375 17.28125 19.8125 11.734375 \n",
       "Q 24.65625 6.203125 33.40625 6.203125 \n",
       "Q 37.3125 6.203125 41.140625 7.25 \n",
       "Q 44.96875 8.296875 48.78125 10.40625 \n",
       "L 48.78125 2.09375 \n",
       "Q 45.015625 0.34375 40.984375 -0.53125 \n",
       "Q 36.96875 -1.421875 32.421875 -1.421875 \n",
       "Q 20.0625 -1.421875 12.78125 6.34375 \n",
       "Q 5.515625 14.109375 5.515625 27.296875 \n",
       "Q 5.515625 40.671875 12.859375 48.328125 \n",
       "Q 20.21875 56 33.015625 56 \n",
       "Q 37.15625 56 41.109375 55.140625 \n",
       "Q 45.0625 54.296875 48.78125 52.59375 \n",
       "z\n",
       "\" id=\"DejaVuSans-99\"/>\n",
       "       <path d=\"M 18.3125 70.21875 \n",
       "L 18.3125 54.6875 \n",
       "L 36.8125 54.6875 \n",
       "L 36.8125 47.703125 \n",
       "L 18.3125 47.703125 \n",
       "L 18.3125 18.015625 \n",
       "Q 18.3125 11.328125 20.140625 9.421875 \n",
       "Q 21.96875 7.515625 27.59375 7.515625 \n",
       "L 36.8125 7.515625 \n",
       "L 36.8125 0 \n",
       "L 27.59375 0 \n",
       "Q 17.1875 0 13.234375 3.875 \n",
       "Q 9.28125 7.765625 9.28125 18.015625 \n",
       "L 9.28125 47.703125 \n",
       "L 2.6875 47.703125 \n",
       "L 2.6875 54.6875 \n",
       "L 9.28125 54.6875 \n",
       "L 9.28125 70.21875 \n",
       "z\n",
       "\" id=\"DejaVuSans-116\"/>\n",
       "       <path d=\"M 16.796875 72.90625 \n",
       "L 26.703125 72.90625 \n",
       "L 14.203125 8.203125 \n",
       "L 49.703125 8.203125 \n",
       "L 48.09375 0 \n",
       "L 2.6875 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-Oblique-76\"/>\n",
       "      </defs>\n",
       "      <use transform=\"translate(0 0.015625)\" xlink:href=\"#DejaVuSans-111\"/>\n",
       "      <use transform=\"translate(61.181641 0.015625)\" xlink:href=\"#DejaVuSans-118\"/>\n",
       "      <use transform=\"translate(120.361328 0.015625)\" xlink:href=\"#DejaVuSans-101\"/>\n",
       "      <use transform=\"translate(181.884766 0.015625)\" xlink:href=\"#DejaVuSans-114\"/>\n",
       "      <use transform=\"translate(222.998047 0.015625)\" xlink:href=\"#DejaVuSans-115\"/>\n",
       "      <use transform=\"translate(275.097656 0.015625)\" xlink:href=\"#DejaVuSans-97\"/>\n",
       "      <use transform=\"translate(336.376953 0.015625)\" xlink:href=\"#DejaVuSans-109\"/>\n",
       "      <use transform=\"translate(433.789062 0.015625)\" xlink:href=\"#DejaVuSans-112\"/>\n",
       "      <use transform=\"translate(497.265625 0.015625)\" xlink:href=\"#DejaVuSans-108\"/>\n",
       "      <use transform=\"translate(525.048828 0.015625)\" xlink:href=\"#DejaVuSans-105\"/>\n",
       "      <use transform=\"translate(552.832031 0.015625)\" xlink:href=\"#DejaVuSans-110\"/>\n",
       "      <use transform=\"translate(616.210938 0.015625)\" xlink:href=\"#DejaVuSans-103\"/>\n",
       "      <use transform=\"translate(679.6875 0.015625)\" xlink:href=\"#DejaVuSans-32\"/>\n",
       "      <use transform=\"translate(711.474609 0.015625)\" xlink:href=\"#DejaVuSans-102\"/>\n",
       "      <use transform=\"translate(746.679688 0.015625)\" xlink:href=\"#DejaVuSans-97\"/>\n",
       "      <use transform=\"translate(807.958984 0.015625)\" xlink:href=\"#DejaVuSans-99\"/>\n",
       "      <use transform=\"translate(862.939453 0.015625)\" xlink:href=\"#DejaVuSans-116\"/>\n",
       "      <use transform=\"translate(902.148438 0.015625)\" xlink:href=\"#DejaVuSans-111\"/>\n",
       "      <use transform=\"translate(963.330078 0.015625)\" xlink:href=\"#DejaVuSans-114\"/>\n",
       "      <use transform=\"translate(1004.443359 0.015625)\" xlink:href=\"#DejaVuSans-32\"/>\n",
       "      <use transform=\"translate(1036.230469 0.015625)\" xlink:href=\"#DejaVuSans-Oblique-76\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"matplotlib.axis_2\">\n",
       "    <g id=\"ytick_1\">\n",
       "     <g id=\"line2d_26\">\n",
       "      <path clip-path=\"url(#p8f4f71aab9)\" d=\"M 46.965625 205.164984 \n",
       "L 604.965625 205.164984 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_27\">\n",
       "      <defs>\n",
       "       <path d=\"M 0 0 \n",
       "L -3.5 0 \n",
       "\" id=\"me6f828898d\" style=\"stroke:#000000;stroke-width:0.8;\"/>\n",
       "      </defs>\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"46.965625\" xlink:href=\"#me6f828898d\" y=\"205.164984\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_4\">\n",
       "      <!-- 100 -->\n",
       "      <g transform=\"translate(20.878125 208.964203)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"127.246094\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_2\">\n",
       "     <g id=\"line2d_28\">\n",
       "      <path clip-path=\"url(#p8f4f71aab9)\" d=\"M 46.965625 169.448796 \n",
       "L 604.965625 169.448796 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_29\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"46.965625\" xlink:href=\"#me6f828898d\" y=\"169.448796\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_5\">\n",
       "      <!-- 105 -->\n",
       "      <g transform=\"translate(20.878125 173.248014)scale(0.1 -0.1)\">\n",
       "       <defs>\n",
       "        <path d=\"M 10.796875 72.90625 \n",
       "L 49.515625 72.90625 \n",
       "L 49.515625 64.59375 \n",
       "L 19.828125 64.59375 \n",
       "L 19.828125 46.734375 \n",
       "Q 21.96875 47.46875 24.109375 47.828125 \n",
       "Q 26.265625 48.1875 28.421875 48.1875 \n",
       "Q 40.625 48.1875 47.75 41.5 \n",
       "Q 54.890625 34.8125 54.890625 23.390625 \n",
       "Q 54.890625 11.625 47.5625 5.09375 \n",
       "Q 40.234375 -1.421875 26.90625 -1.421875 \n",
       "Q 22.3125 -1.421875 17.546875 -0.640625 \n",
       "Q 12.796875 0.140625 7.71875 1.703125 \n",
       "L 7.71875 11.625 \n",
       "Q 12.109375 9.234375 16.796875 8.0625 \n",
       "Q 21.484375 6.890625 26.703125 6.890625 \n",
       "Q 35.15625 6.890625 40.078125 11.328125 \n",
       "Q 45.015625 15.765625 45.015625 23.390625 \n",
       "Q 45.015625 31 40.078125 35.4375 \n",
       "Q 35.15625 39.890625 26.703125 39.890625 \n",
       "Q 22.75 39.890625 18.8125 39.015625 \n",
       "Q 14.890625 38.140625 10.796875 36.28125 \n",
       "z\n",
       "\" id=\"DejaVuSans-53\"/>\n",
       "       </defs>\n",
       "       <use xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "       <use x=\"127.246094\" xlink:href=\"#DejaVuSans-53\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_3\">\n",
       "     <g id=\"line2d_30\">\n",
       "      <path clip-path=\"url(#p8f4f71aab9)\" d=\"M 46.965625 133.732607 \n",
       "L 604.965625 133.732607 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_31\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"46.965625\" xlink:href=\"#me6f828898d\" y=\"133.732607\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_6\">\n",
       "      <!-- 110 -->\n",
       "      <g transform=\"translate(20.878125 137.531826)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"127.246094\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_4\">\n",
       "     <g id=\"line2d_32\">\n",
       "      <path clip-path=\"url(#p8f4f71aab9)\" d=\"M 46.965625 98.016419 \n",
       "L 604.965625 98.016419 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_33\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"46.965625\" xlink:href=\"#me6f828898d\" y=\"98.016419\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_7\">\n",
       "      <!-- 115 -->\n",
       "      <g transform=\"translate(20.878125 101.815638)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"127.246094\" xlink:href=\"#DejaVuSans-53\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_5\">\n",
       "     <g id=\"line2d_34\">\n",
       "      <path clip-path=\"url(#p8f4f71aab9)\" d=\"M 46.965625 62.300231 \n",
       "L 604.965625 62.300231 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_35\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"46.965625\" xlink:href=\"#me6f828898d\" y=\"62.300231\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_8\">\n",
       "      <!-- 120 -->\n",
       "      <g transform=\"translate(20.878125 66.09945)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-50\"/>\n",
       "       <use x=\"127.246094\" xlink:href=\"#DejaVuSans-48\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"ytick_6\">\n",
       "     <g id=\"line2d_36\">\n",
       "      <path clip-path=\"url(#p8f4f71aab9)\" d=\"M 46.965625 26.584043 \n",
       "L 604.965625 26.584043 \n",
       "\" style=\"fill:none;stroke:#b0b0b0;stroke-linecap:square;stroke-width:0.8;\"/>\n",
       "     </g>\n",
       "     <g id=\"line2d_37\">\n",
       "      <g>\n",
       "       <use style=\"stroke:#000000;stroke-width:0.8;\" x=\"46.965625\" xlink:href=\"#me6f828898d\" y=\"26.584043\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "     <g id=\"text_9\">\n",
       "      <!-- 125 -->\n",
       "      <g transform=\"translate(20.878125 30.383262)scale(0.1 -0.1)\">\n",
       "       <use xlink:href=\"#DejaVuSans-49\"/>\n",
       "       <use x=\"63.623047\" xlink:href=\"#DejaVuSans-50\"/>\n",
       "       <use x=\"127.246094\" xlink:href=\"#DejaVuSans-53\"/>\n",
       "      </g>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"text_10\">\n",
       "     <!-- SNR in dB -->\n",
       "     <g transform=\"translate(14.798437 142.809687)rotate(-90)scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path d=\"M 53.515625 70.515625 \n",
       "L 53.515625 60.890625 \n",
       "Q 47.90625 63.578125 42.921875 64.890625 \n",
       "Q 37.9375 66.21875 33.296875 66.21875 \n",
       "Q 25.25 66.21875 20.875 63.09375 \n",
       "Q 16.5 59.96875 16.5 54.203125 \n",
       "Q 16.5 49.359375 19.40625 46.890625 \n",
       "Q 22.3125 44.4375 30.421875 42.921875 \n",
       "L 36.375 41.703125 \n",
       "Q 47.40625 39.59375 52.65625 34.296875 \n",
       "Q 57.90625 29 57.90625 20.125 \n",
       "Q 57.90625 9.515625 50.796875 4.046875 \n",
       "Q 43.703125 -1.421875 29.984375 -1.421875 \n",
       "Q 24.8125 -1.421875 18.96875 -0.25 \n",
       "Q 13.140625 0.921875 6.890625 3.21875 \n",
       "L 6.890625 13.375 \n",
       "Q 12.890625 10.015625 18.65625 8.296875 \n",
       "Q 24.421875 6.59375 29.984375 6.59375 \n",
       "Q 38.421875 6.59375 43.015625 9.90625 \n",
       "Q 47.609375 13.234375 47.609375 19.390625 \n",
       "Q 47.609375 24.75 44.3125 27.78125 \n",
       "Q 41.015625 30.8125 33.5 32.328125 \n",
       "L 27.484375 33.5 \n",
       "Q 16.453125 35.6875 11.515625 40.375 \n",
       "Q 6.59375 45.0625 6.59375 53.421875 \n",
       "Q 6.59375 63.09375 13.40625 68.65625 \n",
       "Q 20.21875 74.21875 32.171875 74.21875 \n",
       "Q 37.3125 74.21875 42.625 73.28125 \n",
       "Q 47.953125 72.359375 53.515625 70.515625 \n",
       "z\n",
       "\" id=\"DejaVuSans-83\"/>\n",
       "       <path d=\"M 9.8125 72.90625 \n",
       "L 23.09375 72.90625 \n",
       "L 55.421875 11.921875 \n",
       "L 55.421875 72.90625 \n",
       "L 64.984375 72.90625 \n",
       "L 64.984375 0 \n",
       "L 51.703125 0 \n",
       "L 19.390625 60.984375 \n",
       "L 19.390625 0 \n",
       "L 9.8125 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-78\"/>\n",
       "       <path d=\"M 44.390625 34.1875 \n",
       "Q 47.5625 33.109375 50.5625 29.59375 \n",
       "Q 53.5625 26.078125 56.59375 19.921875 \n",
       "L 66.609375 0 \n",
       "L 56 0 \n",
       "L 46.6875 18.703125 \n",
       "Q 43.0625 26.03125 39.671875 28.421875 \n",
       "Q 36.28125 30.8125 30.421875 30.8125 \n",
       "L 19.671875 30.8125 \n",
       "L 19.671875 0 \n",
       "L 9.8125 0 \n",
       "L 9.8125 72.90625 \n",
       "L 32.078125 72.90625 \n",
       "Q 44.578125 72.90625 50.734375 67.671875 \n",
       "Q 56.890625 62.453125 56.890625 51.90625 \n",
       "Q 56.890625 45.015625 53.6875 40.46875 \n",
       "Q 50.484375 35.9375 44.390625 34.1875 \n",
       "z\n",
       "M 19.671875 64.796875 \n",
       "L 19.671875 38.921875 \n",
       "L 32.078125 38.921875 \n",
       "Q 39.203125 38.921875 42.84375 42.21875 \n",
       "Q 46.484375 45.515625 46.484375 51.90625 \n",
       "Q 46.484375 58.296875 42.84375 61.546875 \n",
       "Q 39.203125 64.796875 32.078125 64.796875 \n",
       "z\n",
       "\" id=\"DejaVuSans-82\"/>\n",
       "       <path d=\"M 45.40625 46.390625 \n",
       "L 45.40625 75.984375 \n",
       "L 54.390625 75.984375 \n",
       "L 54.390625 0 \n",
       "L 45.40625 0 \n",
       "L 45.40625 8.203125 \n",
       "Q 42.578125 3.328125 38.25 0.953125 \n",
       "Q 33.9375 -1.421875 27.875 -1.421875 \n",
       "Q 17.96875 -1.421875 11.734375 6.484375 \n",
       "Q 5.515625 14.40625 5.515625 27.296875 \n",
       "Q 5.515625 40.1875 11.734375 48.09375 \n",
       "Q 17.96875 56 27.875 56 \n",
       "Q 33.9375 56 38.25 53.625 \n",
       "Q 42.578125 51.265625 45.40625 46.390625 \n",
       "z\n",
       "M 14.796875 27.296875 \n",
       "Q 14.796875 17.390625 18.875 11.75 \n",
       "Q 22.953125 6.109375 30.078125 6.109375 \n",
       "Q 37.203125 6.109375 41.296875 11.75 \n",
       "Q 45.40625 17.390625 45.40625 27.296875 \n",
       "Q 45.40625 37.203125 41.296875 42.84375 \n",
       "Q 37.203125 48.484375 30.078125 48.484375 \n",
       "Q 22.953125 48.484375 18.875 42.84375 \n",
       "Q 14.796875 37.203125 14.796875 27.296875 \n",
       "z\n",
       "\" id=\"DejaVuSans-100\"/>\n",
       "       <path d=\"M 19.671875 34.8125 \n",
       "L 19.671875 8.109375 \n",
       "L 35.5 8.109375 \n",
       "Q 43.453125 8.109375 47.28125 11.40625 \n",
       "Q 51.125 14.703125 51.125 21.484375 \n",
       "Q 51.125 28.328125 47.28125 31.5625 \n",
       "Q 43.453125 34.8125 35.5 34.8125 \n",
       "z\n",
       "M 19.671875 64.796875 \n",
       "L 19.671875 42.828125 \n",
       "L 34.28125 42.828125 \n",
       "Q 41.5 42.828125 45.03125 45.53125 \n",
       "Q 48.578125 48.25 48.578125 53.8125 \n",
       "Q 48.578125 59.328125 45.03125 62.0625 \n",
       "Q 41.5 64.796875 34.28125 64.796875 \n",
       "z\n",
       "M 9.8125 72.90625 \n",
       "L 35.015625 72.90625 \n",
       "Q 46.296875 72.90625 52.390625 68.21875 \n",
       "Q 58.5 63.53125 58.5 54.890625 \n",
       "Q 58.5 48.1875 55.375 44.234375 \n",
       "Q 52.25 40.28125 46.1875 39.3125 \n",
       "Q 53.46875 37.75 57.5 32.78125 \n",
       "Q 61.53125 27.828125 61.53125 20.40625 \n",
       "Q 61.53125 10.640625 54.890625 5.3125 \n",
       "Q 48.25 0 35.984375 0 \n",
       "L 9.8125 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-66\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-83\"/>\n",
       "      <use x=\"63.476562\" xlink:href=\"#DejaVuSans-78\"/>\n",
       "      <use x=\"138.28125\" xlink:href=\"#DejaVuSans-82\"/>\n",
       "      <use x=\"207.763672\" xlink:href=\"#DejaVuSans-32\"/>\n",
       "      <use x=\"239.550781\" xlink:href=\"#DejaVuSans-105\"/>\n",
       "      <use x=\"267.333984\" xlink:href=\"#DejaVuSans-110\"/>\n",
       "      <use x=\"330.712891\" xlink:href=\"#DejaVuSans-32\"/>\n",
       "      <use x=\"362.5\" xlink:href=\"#DejaVuSans-100\"/>\n",
       "      <use x=\"425.976562\" xlink:href=\"#DejaVuSans-66\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "   <g id=\"line2d_38\">\n",
       "    <path clip-path=\"url(#p8f4f71aab9)\" d=\"M 72.329261 199.541207 \n",
       "L 135.738352 176.112674 \n",
       "L 199.147443 154.736978 \n",
       "L 262.556534 134.155325 \n",
       "L 325.965625 111.068199 \n",
       "L 389.374716 90.939658 \n",
       "L 452.783807 66.376386 \n",
       "L 516.192898 42.022724 \n",
       "L 579.601989 17.28 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-linecap:square;stroke-width:1.5;\"/>\n",
       "   </g>\n",
       "   <g id=\"line2d_39\">\n",
       "    <path clip-path=\"url(#p8f4f71aab9)\" d=\"M 72.329261 218.88 \n",
       "L 135.738352 218.88 \n",
       "L 199.147443 218.88 \n",
       "L 262.556534 218.88 \n",
       "L 325.965625 218.88 \n",
       "L 389.374716 218.88 \n",
       "L 452.783807 218.88 \n",
       "L 516.192898 218.88 \n",
       "L 579.601989 218.88 \n",
       "\" style=\"fill:none;stroke:#ff7f0e;stroke-linecap:square;stroke-width:1.5;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_3\">\n",
       "    <path d=\"M 46.965625 228.96 \n",
       "L 46.965625 7.2 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_4\">\n",
       "    <path d=\"M 604.965625 228.96 \n",
       "L 604.965625 7.2 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_5\">\n",
       "    <path d=\"M 46.965625 228.96 \n",
       "L 604.965625 228.96 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "   <g id=\"patch_6\">\n",
       "    <path d=\"M 46.965625 7.2 \n",
       "L 604.965625 7.2 \n",
       "\" style=\"fill:none;stroke:#000000;stroke-linecap:square;stroke-linejoin:miter;stroke-width:0.8;\"/>\n",
       "   </g>\n",
       "   <g id=\"legend_1\">\n",
       "    <g id=\"patch_7\">\n",
       "     <path d=\"M 53.965625 44.55625 \n",
       "L 202.284375 44.55625 \n",
       "Q 204.284375 44.55625 204.284375 42.55625 \n",
       "L 204.284375 14.2 \n",
       "Q 204.284375 12.2 202.284375 12.2 \n",
       "L 53.965625 12.2 \n",
       "Q 51.965625 12.2 51.965625 14.2 \n",
       "L 51.965625 42.55625 \n",
       "Q 51.965625 44.55625 53.965625 44.55625 \n",
       "z\n",
       "\" style=\"fill:#ffffff;opacity:0.8;stroke:#cccccc;stroke-linejoin:miter;\"/>\n",
       "    </g>\n",
       "    <g id=\"line2d_40\">\n",
       "     <path d=\"M 55.965625 20.298437 \n",
       "L 75.965625 20.298437 \n",
       "\" style=\"fill:none;stroke:#1f77b4;stroke-linecap:square;stroke-width:1.5;\"/>\n",
       "    </g>\n",
       "    <g id=\"line2d_41\"/>\n",
       "    <g id=\"text_11\">\n",
       "     <!-- SNR with oversampling -->\n",
       "     <g transform=\"translate(83.965625 23.798437)scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path d=\"M 4.203125 54.6875 \n",
       "L 13.1875 54.6875 \n",
       "L 24.421875 12.015625 \n",
       "L 35.59375 54.6875 \n",
       "L 46.1875 54.6875 \n",
       "L 57.421875 12.015625 \n",
       "L 68.609375 54.6875 \n",
       "L 77.59375 54.6875 \n",
       "L 63.28125 0 \n",
       "L 52.6875 0 \n",
       "L 40.921875 44.828125 \n",
       "L 29.109375 0 \n",
       "L 18.5 0 \n",
       "z\n",
       "\" id=\"DejaVuSans-119\"/>\n",
       "       <path d=\"M 54.890625 33.015625 \n",
       "L 54.890625 0 \n",
       "L 45.90625 0 \n",
       "L 45.90625 32.71875 \n",
       "Q 45.90625 40.484375 42.875 44.328125 \n",
       "Q 39.84375 48.1875 33.796875 48.1875 \n",
       "Q 26.515625 48.1875 22.3125 43.546875 \n",
       "Q 18.109375 38.921875 18.109375 30.90625 \n",
       "L 18.109375 0 \n",
       "L 9.078125 0 \n",
       "L 9.078125 75.984375 \n",
       "L 18.109375 75.984375 \n",
       "L 18.109375 46.1875 \n",
       "Q 21.34375 51.125 25.703125 53.5625 \n",
       "Q 30.078125 56 35.796875 56 \n",
       "Q 45.21875 56 50.046875 50.171875 \n",
       "Q 54.890625 44.34375 54.890625 33.015625 \n",
       "z\n",
       "\" id=\"DejaVuSans-104\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-83\"/>\n",
       "      <use x=\"63.476562\" xlink:href=\"#DejaVuSans-78\"/>\n",
       "      <use x=\"138.28125\" xlink:href=\"#DejaVuSans-82\"/>\n",
       "      <use x=\"207.763672\" xlink:href=\"#DejaVuSans-32\"/>\n",
       "      <use x=\"239.550781\" xlink:href=\"#DejaVuSans-119\"/>\n",
       "      <use x=\"321.337891\" xlink:href=\"#DejaVuSans-105\"/>\n",
       "      <use x=\"349.121094\" xlink:href=\"#DejaVuSans-116\"/>\n",
       "      <use x=\"388.330078\" xlink:href=\"#DejaVuSans-104\"/>\n",
       "      <use x=\"451.708984\" xlink:href=\"#DejaVuSans-32\"/>\n",
       "      <use x=\"483.496094\" xlink:href=\"#DejaVuSans-111\"/>\n",
       "      <use x=\"544.677734\" xlink:href=\"#DejaVuSans-118\"/>\n",
       "      <use x=\"603.857422\" xlink:href=\"#DejaVuSans-101\"/>\n",
       "      <use x=\"665.380859\" xlink:href=\"#DejaVuSans-114\"/>\n",
       "      <use x=\"706.494141\" xlink:href=\"#DejaVuSans-115\"/>\n",
       "      <use x=\"758.59375\" xlink:href=\"#DejaVuSans-97\"/>\n",
       "      <use x=\"819.873047\" xlink:href=\"#DejaVuSans-109\"/>\n",
       "      <use x=\"917.285156\" xlink:href=\"#DejaVuSans-112\"/>\n",
       "      <use x=\"980.761719\" xlink:href=\"#DejaVuSans-108\"/>\n",
       "      <use x=\"1008.544922\" xlink:href=\"#DejaVuSans-105\"/>\n",
       "      <use x=\"1036.328125\" xlink:href=\"#DejaVuSans-110\"/>\n",
       "      <use x=\"1099.707031\" xlink:href=\"#DejaVuSans-103\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "    <g id=\"line2d_42\">\n",
       "     <path d=\"M 55.965625 34.976562 \n",
       "L 75.965625 34.976562 \n",
       "\" style=\"fill:none;stroke:#ff7f0e;stroke-linecap:square;stroke-width:1.5;\"/>\n",
       "    </g>\n",
       "    <g id=\"line2d_43\"/>\n",
       "    <g id=\"text_12\">\n",
       "     <!-- SNR w/o oversampling -->\n",
       "     <g transform=\"translate(83.965625 38.476562)scale(0.1 -0.1)\">\n",
       "      <defs>\n",
       "       <path d=\"M 25.390625 72.90625 \n",
       "L 33.6875 72.90625 \n",
       "L 8.296875 -9.28125 \n",
       "L 0 -9.28125 \n",
       "z\n",
       "\" id=\"DejaVuSans-47\"/>\n",
       "      </defs>\n",
       "      <use xlink:href=\"#DejaVuSans-83\"/>\n",
       "      <use x=\"63.476562\" xlink:href=\"#DejaVuSans-78\"/>\n",
       "      <use x=\"138.28125\" xlink:href=\"#DejaVuSans-82\"/>\n",
       "      <use x=\"207.763672\" xlink:href=\"#DejaVuSans-32\"/>\n",
       "      <use x=\"239.550781\" xlink:href=\"#DejaVuSans-119\"/>\n",
       "      <use x=\"321.337891\" xlink:href=\"#DejaVuSans-47\"/>\n",
       "      <use x=\"355.029297\" xlink:href=\"#DejaVuSans-111\"/>\n",
       "      <use x=\"416.210938\" xlink:href=\"#DejaVuSans-32\"/>\n",
       "      <use x=\"447.998047\" xlink:href=\"#DejaVuSans-111\"/>\n",
       "      <use x=\"509.179688\" xlink:href=\"#DejaVuSans-118\"/>\n",
       "      <use x=\"568.359375\" xlink:href=\"#DejaVuSans-101\"/>\n",
       "      <use x=\"629.882812\" xlink:href=\"#DejaVuSans-114\"/>\n",
       "      <use x=\"670.996094\" xlink:href=\"#DejaVuSans-115\"/>\n",
       "      <use x=\"723.095703\" xlink:href=\"#DejaVuSans-97\"/>\n",
       "      <use x=\"784.375\" xlink:href=\"#DejaVuSans-109\"/>\n",
       "      <use x=\"881.787109\" xlink:href=\"#DejaVuSans-112\"/>\n",
       "      <use x=\"945.263672\" xlink:href=\"#DejaVuSans-108\"/>\n",
       "      <use x=\"973.046875\" xlink:href=\"#DejaVuSans-105\"/>\n",
       "      <use x=\"1000.830078\" xlink:href=\"#DejaVuSans-110\"/>\n",
       "      <use x=\"1064.208984\" xlink:href=\"#DejaVuSans-103\"/>\n",
       "     </g>\n",
       "    </g>\n",
       "   </g>\n",
       "  </g>\n",
       " </g>\n",
       " <defs>\n",
       "  <clipPath id=\"p8f4f71aab9\">\n",
       "   <rect height=\"221.76\" width=\"558\" x=\"46.965625\" y=\"7.2\"/>\n",
       "  </clipPath>\n",
       " </defs>\n",
       "</svg>\n"
      ],
      "text/plain": [
       "<Figure size 960x384 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import scipy.signal as sig\n",
    "\n",
    "w = 16  # wordlength of the quantized signal\n",
    "L = 2 ** np.arange(1, 10)  # oversampling factors\n",
    "\n",
    "N = 8192  # length of signal\n",
    "Om0 = 100 * 2 * np.pi / N  # frequency of harmonic signal\n",
    "Q = 1 / (2 ** (w - 1))  # quantization step\n",
    "\n",
    "\n",
    "def uniform_midtread_quantizer(x, Q):\n",
    "    \"\"\"Uniform mid-tread qantizer with limiter.\"\"\"\n",
    "    # limiter\n",
    "    x = np.copy(x)\n",
    "    idx = np.where(x <= -1)\n",
    "    x[idx] = -1\n",
    "    idx = np.where(x > 1 - Q)\n",
    "    x[idx] = 1 - Q\n",
    "    # linear uniform quantization\n",
    "    xQ = Q * np.floor(x / Q + 1 / 2)\n",
    "\n",
    "    return xQ\n",
    "\n",
    "\n",
    "def SNR_oversampled_ADC(L):\n",
    "    \"\"\"Estimate SNR of oversampled analog-to-digital converter.\"\"\"\n",
    "    x = (1 - Q) * np.cos(Om0 * np.arange(N))\n",
    "    xu = (1 - Q) * np.cos(Om0 * np.arange(N * L) / L)\n",
    "    # quantize signal\n",
    "    xQu = uniform_midtread_quantizer(xu, Q)\n",
    "    # low-pass filtering and decimation\n",
    "    xQ = sig.resample(xQu, N)\n",
    "    # estimate SNR\n",
    "    e = xQ - x\n",
    "\n",
    "    return 10 * np.log10((np.var(x) / np.var(e)))\n",
    "\n",
    "\n",
    "# compute SNR for oversampled ADC\n",
    "SNR = [SNR_oversampled_ADC(l) for l in L]\n",
    "\n",
    "# plot result\n",
    "plt.figure(figsize=(10, 4))\n",
    "plt.semilogx(L, SNR, label=\"SNR with oversampling\")\n",
    "plt.plot(L, (6.02 * w + 1.76) * np.ones(L.shape), label=\"SNR w/o oversampling\")\n",
    "plt.xlabel(r\"oversampling factor $L$\")\n",
    "plt.ylabel(r\"SNR in dB\")\n",
    "plt.legend(loc=\"upper left\")\n",
    "plt.grid()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**\n",
    "\n",
    "* What SNR can be achieved for an oversampling factor of $L=16$?\n",
    "* By how many bits could the word length $w$ be reduced in order to gain the same SNR as without oversampling?\n",
    "\n",
    "Solution: The SNR for the quantization of a uniformly distributed input signal without oversampling is $\\text{SNR} \\approx 6.02 w \\approx 96$ dB and with 16 times oversampling $\\text{SNR}_{L} \\approx 6.02 w + 10 \\cdot \\log_{10} (16)  \\approx 96 + 12$ dB. Since the [quantization of a harmonic signal](linear_uniform_quantization_error.ipynb#Harmonic-Signal) is considered an offset of $1.76$ dB has to added to both. The wordlength could be reduced by 2 bits according to these numbers."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Anti-Aliasing Filter\n",
    "\n",
    "Besides an increased SNR, oversampling has also another benefit. In order to ensure that the input signal $x(t)$ is band-limited before sampling, a low-pass filter $H_\\text{LP}(\\mathrm{j}\\,\\omega)$ is applied in typical analog-to-digital converters. This is illustrated in the following\n",
    "\n",
    "![Oversampling ideal analog to digital converter with anti-aliasing filter](oversampling_anti_aliasing.png)\n",
    "\n",
    "The filter $H_\\text{LP}(\\mathrm{j}\\,\\omega)$ is also known as [anti-aliasing filter](https://en.wikipedia.org/wiki/Anti-aliasing_filter). The ideal low-pass filter is given as $H_\\text{LP}(\\mathrm{j}\\,\\omega) = \\text{rect}\\left( \\frac{\\omega}{\\omega_\\text{S}}  \\right)$. The ideal $H_\\text{LP}(\\mathrm{j}\\,\\omega)$ can only be approximated in the analog domain. Since the sampling rate is higher than the Nyquist rate, there is no need for a steep slope of the filter in order to avoid aliasing. However, the pass-band of the filter within $|\\omega| < |\\omega_\\text{C}|$ has to be flat. \n",
    "\n",
    "Before decimation, the discrete filter $H_\\text{LP}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ has to remove the spectral contributions that may lead to aliasing. However, a discrete filter $H_\\text{LP}(\\mathrm{e}^{\\,\\mathrm{j}\\,\\Omega})$ with steep slope can be realized much easier than in the analog domain."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Digital Signal Processing - Lecture notes featuring computational examples*."
   ]
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.7.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}
